Define \(Z_t^{(i)} = X_t \cdot \mathbb {I}\left\lbrace U_t \le \exp (i)\right\rbrace \) , then \(Z_t^{(i)}\exp (-i)\le 1\)  almost surely.
By Exercise 5.15 of [1]}, with probability at least \(1-\delta /(2i^2)\) , we have
\(\sum _{t=1}^T Z_t^{(i)} \le \sum _{t=1}^T (Z_t^{(i)}-\mathbb {E}[Z_t^{(i)}~|~\mathcal {F}_t])\le \exp (-i)\sum _{t=1}^T\mathbb {E}\left[\left(Z_t^{(i)}\right)^2~\Big |~\mathcal {F}_t\right]+\exp (i)\ln \left(\frac{2 i^2}{\delta }\right)\,.\) 
